If x, y and z are positive numbers, is x/y (x + z)/(y + z) ?

Question

If x, y and z are positive numbers, is  \frac{x}{y} > \frac{x + z}{y + z}  ?

(1) x > z

(2) x > y

M36-64

Bunuel · Accepted Answer

Official Solution:

If  x ,  y  and  z  are positive numbers, is  \frac{x}{y} >\frac{x + z}{y + z}  ? 
 
Since the unknowns are positive we can safely cross-multiply and simplify:
 
Is  x(y+z) >y(x + z) ? 
 
Is  xy+xz >yx + yz ? 
 
Is  xz >yz ? 
 
Reduce by  z : Is  x >y ? 
 
(1)  x > z . Not sufficient.
 
(2)  x > y . Sufficient.

Answer: B

vaibhav1221 · Answer

\frac{x}{y} > \frac{x + z}{y + z} 
Cross multiplying
xy + xz > xy + yz
  xy   + x  z   >   xy   + y  z  
x>y

Hence, B

Archit3110 · Answer

From given eqn we can deduce the following:
xy+xz>xy+yz
z(x-y)>0--( A )

so using 1: x>z , we are not given any relation about y so in -sufficient

From 1: X>Y 
for any value of X >Y value the equation (  A)  would stand true so B is sufficient..

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

GMATin · Answer

There are two methods to solve this question.

Method 1:

Since x, y, z are all positive numbers, we can easily carry their denominators over the inequality without the fear of messing with the inequality.

Hence, \(\frac{x}{y} > \frac{x + z}{y + z}\) can be written as x(y+z)>y(x+z)
This implies xy+xz>xy+yz

Again, xy is a positive number here, so we cancel it out from both sides, leaving us with xz>yz. Taking the expression to one side of inequality, we get xz-yz>0

Thus, either z>0 and x>y or z<0 and x<y.

Now, we are given that z>0 (its a positive number). Hence, for the expression to be definitely true, we must also get x>y. And statement 2 does that for us.

Hence statement 2 is sufficient

Method 2:

Now, if we add the same positive number z to x and y in the fraction x/y, \(\frac{x + z}{y + z}\) will come closer to 1.

This gives us two possibilities:

1) If x/y<1, then \(\frac{(x+z)}{(y+z)}\) is greater than x/y

and

2) If x/y>1, then \(\frac{(x+z)}{(y+z)}\) is smaller than x/y

In our case, 2) works and this is satisfied only by statement 2. Hence, sufficient

Answer is B

Note: For anyone looking for more explanation on concept I used in statement 2, there is an article I think written by Karishma somewhere on GMATclub explaining this. If interested, I can look it up.

mbaORnoMBA · Answer

For x/y to be greater, x/y needs to be greater than 1 or x>y

S1 irrelevant
S2 x>y hence a definitive Yes -> sufficient

If x, y and z are positive numbers, is x/y > (x + z)/(y + z) ?

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